Twin Paradox

I'm trying to understand how time will be observed to have stopped for the traveller from an observer on earth when a traveller travels at light speed, makes a round trip and returns to earth.

Would this thinking be correct?

The traveller is travelling infinitely close to the speed of light, since he makes a round trip (changing direction and returning home), the total time for the traveller will be less than the time elapsed on earth, therefore causing the traveller to have aged less than the observer.

The closer to the speed of light the traveller travels at, the less time elapses for the traveller in one round trip, until the time elapsed is infinitely small, and to an observer who has aged any amount of time (depending on how many round trips the traveller makes) it appears that no time has elapsed for the traveller.

There's no such thing as "infinitely small" (at least for ordinary numbers). It is theoretically possible to travel at any speed less than the speed of light, but it is impossible to travel at or above the speed of light. A twin travelling incredibly close to the speed of light can return to earth in as little time as he'd like (again theoretically), and have earth aged as much as he wants, but he can never return in zero time.

You can't actually travel at the speed of light, though you can arbitrarily close. So you can say that the trip approaches zero time as your velocity approaches 'c'. But you can't really say that you can travel 'at c', nor can you say that you actually make the trip in zero time, though you can (in theory) make the trip in an arbitrarily short time.

Twin A is on Earth.
Twin B is on a spaceship. Travels to the nearest star at .9c, turns around, and comes back.

Now, SR states that Twin B will have aged less than Twin A, right?

Here's my question: To both twins, the opposite twin made the trip, right? So to A, it appeared that B left and came back. To B, it appeared that A left and came back. So why does one age more than the other? Doesn't there have to be an absolute frame in order for this to be true?

Here's my question: To both twins, the opposite twin made the trip, right? So to A, it appeared that B left and came back. To B, it appeared that A left and came back. So why does one age more than the other? Doesn't there have to be an absolute frame in order for this to be true?

Well, that argument is the whole basis of why the twin paradox is called a "paradox", so any book or website that talks about the twin paradox should address it (see this one, for example). Basically the reason the argument doesn't work is because while velocity is relative, acceleration is absolute--the twin that turns around will feel G-forces as he does so while the other will not--and the standard equations of SR such as the equation for time dilation as a function of velocity only work in inertial (non-accelerating) frames. You can analyze the twin paradox from any inertial frame you like, such as the frame where the travelling twin was at rest during the outbound phase of the trip while the twin on Earth was moving (in this frame, when the travelling twin turns around he'll be moving even faster than the Earth on the inbound phase), and it turns out you always get the same answer for their ages at the moment they meet.